Oxford

Plebanski generating functions give a compact encoding of the geometry of self-dual Ricci-flat space-times or hyper-Kahler spaces.  They have applications as generating functions for BPS/DT/Gromov-Witten invariants.  We first show that Plebanski's first fundamental form also provides a generating function for the gravitational MHV amplitude.  We then obtain these Plebanski generating functions from the corresponding twistor spaces as the value of the action of new sigma models for holomorphic curves in twistor space.   
In four-dimensions, perturbations of the hyperk¨ahler structure corresponding to positive helicity gravitons. The sigma model’s perturbation theory gives rise to a sum of tree diagrams for the gravity MHV amplitude observed previously in the literature, and their summation via a matrix tree theorem gives a first-principles derivation of Hodges’ determinant formula directly from general relativity. We generalise the twistor sigma model to higher-degree (defined in the first instance with a cosmological constant), giving a new generating principle for the full tree-level graviton S-matrix in general with or without  cosmological constant.  This is joint work with Tim Adamo and Atul Sharma in https://arxiv.org/abs/2103.16984.  

 Given a solution of the Einstein equations, a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. I will then focus in the talk on recent results showing the locally Lipschitz inextendibility of FLRW models with particle horizons and spherically symmetric weak null singularities. The latter in particular apply to the spherically symmetric spacetimes constructed by Luk and Oh, improving their C^2-formulation of strong cosmic censorship to a locally Lipschitz formulation.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1\approx1.14$.
A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. In joint work with C. Pomerance and T.H. Chan, we study the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In particular we prove that $\tau_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets.

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

The periods of a Calabi-Yau manifold are of interest both to number theorists and to physicists. To a number theorist the primary object of interest is the zeta function. I will explain what this is, and why this is of interest also to physicists. For applications it is important to be able to calculate the local zeta function for many primes p. I will set out a method, adapted from a procedure proposed by Alan Lauder that makes the computation of the zeta function practical, in this sense, and comment on the form of the results. This talk is based largely on the recent paper hepth 2104.07816 and presents joint work with Xenia de la Ossa and Duco van Straten.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…